双层非线性耦合反应扩散系统中复杂Turing斑图

李新政; 白占国; 李燕; 赵昆; 贺亚峰

doi:10.7498/aps.62.220503

摘要

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图.

关键词:

Abstract

The influence of Turing modes in two subsystems on pattern formation is investigated by the two-layer non-linearly coupled Brusselator model. It is found that the coupling coefficient and wave number ratio between two Turing modes take an important role in the pattern formation and pattern selection. The kind of pattern changes from simple pattern to complex one with the increase of wave number ratio. When nonlinear coupling coefficient is smaller than 0.1, the short wave mode in system 1 under the action of instability mode in system 2 can form not only simple pattern (such as simple hexagon and quadrilateral and stripe pattern), but also complex pattern due to the resonance coupling between the two Turing modes (such as honeycomb hexagon and super hexagon and complex black-eye pattern), and the transformation process of pattern from quadrilateral to superlattice pattern is observed for the first time under the specific parameters. When nonlinear coupling coefficient is more than 0.1, the simple patterns such as simple hexagon and stripe pattern are obtained only in system 1, because there is no resonance coupling between the two Turing modes in system 1.

Keywords:

作者及机构信息

1.
河北科技大学理学院, 石家庄 050018;

2.
河北大学物理科学与技术学院, 保定 071002

基金项目: 国家自然科学基金(批准号: 11247242)、国家自然科学基金青年科学基金(批准号: 51201057)和河北科技大学科研基金(批准号: QD201225, QD201226)资助的课题.

Authors and contacts

1.
College of Sciences, Hebei University of Science and Technology, Shijiazhuang 050018, China;

2.
College of Physics Science and Technology, Hebei University, Baoding 071002, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11247242), the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 51201057), and the Foundation of Hebei University of Science and Technology, China (Grant Nos. QD201225, QD201226).

参考文献

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[2]	Berenstein I, Dolnik M, Yang L, Zhabotinsky A M, Epstein I R 2004 Phys. Rev. E 70 046219
[3]	Arbell H, Fineberg J 2002 Phys. Rev. E 65 036224
[4]	Sharpe J P, Ramazza P L, Sungar N, Saunders K 2006 Phys. Rev. Lett. 96 094101
[5]	Bois J S, Jlicher F, Grill S W 2011 Phys. Rev. Lett. 106 028103
[6]	Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[7]	Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2003 Phys. Rev. Lett. 91 058302
[8]	Turing A M 1952 Phil. Trans. R. Soc. London B 237 37
[9]	Dong L F, Fan W L, He Y F, Liu F C, Li S F, Gao R L, Wang L 2006 Phys. Rev. E 73 066206
[10]	Duan X X, Ouyang J T, Zhao X F, He F 2009 Phys. Rev. E 80 016202
[11]	Stollenwerk L, Laven J G, Purwins H G 2007 Phys. Rev. Lett. 98 255001
[12]	Shirafuji T, Kitagawa T, Wakai T, Tachibana K 2003 Appl. Phys. Lett. 83 2309
[13]	Shin J, Raja L L 2007 J. Phys. D: Appl. Phys. 40 3145
[14]	Liu C Z, Brown N, Meenan B J 2006 Appl. Surf. Sci. 252 2297
[15]	Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[16]	Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[17]	Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[18]	Barrio R A, Varea C, Aragón J L, Maini P K 1999 Bull. Math. Biol. 61 483
[19]	Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[20]	Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[21]	Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 60 118201]
[22]	Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[23]	Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[24]	Ni W M, Tang M X 2005 Trans. Amer. Math. Soc. 357 3953
[25]	Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[26]	Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett. 26 024702
[27]	Liu H Y, Yang C Y, Tang G N 2013 Acta Phys. Sin. 62 010505 (in Chinese) [刘海英, 杨翠云, 唐国宁 2013 62 010505]
[28]	Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[29]	Dong L F, Li S F, Liu F, Liu F C, Liu S H, Fan W L 2006 Acta Phys. Sin. 55 362 (in Chinese) [董丽芳, 李树锋, 刘峰, 刘富成, 刘书华, 范伟丽 2006 55 362]
[30]	Dong L F, Yang Y J, Fan W L, Yue H, Wang S, Xiao H 2010 Acta Phys. Sin. 59 1917 (in Chinese) [董丽芳, 杨玉杰, 范伟丽, 岳晗, 王帅, 肖红 2010 59 1917]

施引文献

[1]	Schenk C P, Or-Guil M, Bode M, Purwins H G 1997 Phys. Rev. Lett. 78 3781
[2]	Berenstein I, Dolnik M, Yang L, Zhabotinsky A M, Epstein I R 2004 Phys. Rev. E 70 046219
[3]	Arbell H, Fineberg J 2002 Phys. Rev. E 65 036224
[4]	Sharpe J P, Ramazza P L, Sungar N, Saunders K 2006 Phys. Rev. Lett. 96 094101
[5]	Bois J S, Jlicher F, Grill S W 2011 Phys. Rev. Lett. 106 028103
[6]	Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[7]	Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2003 Phys. Rev. Lett. 91 058302
[8]	Turing A M 1952 Phil. Trans. R. Soc. London B 237 37
[9]	Dong L F, Fan W L, He Y F, Liu F C, Li S F, Gao R L, Wang L 2006 Phys. Rev. E 73 066206
[10]	Duan X X, Ouyang J T, Zhao X F, He F 2009 Phys. Rev. E 80 016202
[11]	Stollenwerk L, Laven J G, Purwins H G 2007 Phys. Rev. Lett. 98 255001
[12]	Shirafuji T, Kitagawa T, Wakai T, Tachibana K 2003 Appl. Phys. Lett. 83 2309
[13]	Shin J, Raja L L 2007 J. Phys. D: Appl. Phys. 40 3145
[14]	Liu C Z, Brown N, Meenan B J 2006 Appl. Surf. Sci. 252 2297
[15]	Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[16]	Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[17]	Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[18]	Barrio R A, Varea C, Aragón J L, Maini P K 1999 Bull. Math. Biol. 61 483
[19]	Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[20]	Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[21]	Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 60 118201]
[22]	Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[23]	Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[24]	Ni W M, Tang M X 2005 Trans. Amer. Math. Soc. 357 3953
[25]	Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[26]	Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett. 26 024702
[27]	Liu H Y, Yang C Y, Tang G N 2013 Acta Phys. Sin. 62 010505 (in Chinese) [刘海英, 杨翠云, 唐国宁 2013 62 010505]
[28]	Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[29]	Dong L F, Li S F, Liu F, Liu F C, Liu S H, Fan W L 2006 Acta Phys. Sin. 55 362 (in Chinese) [董丽芳, 李树锋, 刘峰, 刘富成, 刘书华, 范伟丽 2006 55 362]
[30]	Dong L F, Yang Y J, Fan W L, Yue H, Wang S, Xiao H 2010 Acta Phys. Sin. 59 1917 (in Chinese) [董丽芳, 杨玉杰, 范伟丽, 岳晗, 王帅, 肖红 2010 59 1917]

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